Speed of a Bullet: Deriving Formula with D, T & Theta

In summary, the conversation discusses finding the speed of a bullet that is shot through two cardboard disks attached a distance D apart to a rotating shaft. The formula for the bullet's speed is derived using D, T, and the measured angle theta between the positions of the holes in the two disks. The time it takes for the bullet to traverse the gap can be determined from the angle, and the speed can be calculated by dividing the length of the gap by the time. The conversation also mentions that the bullet penetrated the second disk 45[itex]^\circ[/tex] later than the first disk, and that the disks traveled 1/4 turn in the same time it took the bullet to penetrate both disks.
  • #1
tigerseye
16
0
I've been stuck on this problem forever and I don't know how to do it if you can't use r as the radius.
A bullet is shot through two cardboard disks attached a distance D apart to a shaft turning with a rotational period T (see the attached picture). Derive a formula for the bullet speed v in terms of D, T, and a measured angle theta between the position of the hole in the first disk and that of the hole in the second.
 

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  • #2
You know the time required to make a full revolution and from the angle you can determine the time it takes the bullet to traverse the gap. Find the speed of the bullet by just dividing the length of the gap by the time it takes to cross it.
 
  • #3
I no this sounds stupid but how do you find how long it takes to cross the gap !
 
  • #4
If the bullet penetrated the second disk 45[itex]^\circ[/tex] later than where it penetrated the first disk, then the disks traveled 1/4 turn in the time it took the bullet to penetrate both disks, so...
 
  • #5


I understand your frustration with this problem and I am happy to help you find a solution. The speed of a bullet can be determined by using the formula v = d/t, where v is the velocity, d is the distance traveled, and t is the time it takes to travel that distance. However, in this scenario, we cannot use the distance traveled (d) as it involves the radius which we are not given.

Instead, we can use the rotational period (T) to calculate the distance traveled by the bullet. The distance traveled by the bullet can be represented by the circumference of the circle, which is equal to 2πr, where r is the radius of the circle. However, since we cannot use the radius, we can use the angular displacement (theta) and the distance between the two disks (D) to find the distance traveled.

We can use the formula d = rθ, where d is the distance traveled, r is the radius, and θ is the angular displacement. However, instead of using r, we can use D/2 as the distance between the two disks is equal to the diameter of the circle. Therefore, the formula becomes d = (D/2)θ.

Now, we can substitute this value of d in the original formula v = d/t, to get v = ((D/2)θ)/t. This gives us the formula for the speed of the bullet in terms of D, T, and theta.

In order to solve the problem, you will need to measure the value of theta using the angle between the position of the hole in the first disk and that of the hole in the second. You will also need to measure the distance between the two disks (D) and the rotational period (T). Once you have these values, you can plug them into the formula v = ((D/2)θ)/t to calculate the speed of the bullet.

I hope this explanation helps you solve the problem. Remember, as a scientist, it is important to think critically and use different methods to solve problems when faced with limitations. Keep up the good work!
 

1. What is the formula for calculating the speed of a bullet using distance, time, and angle?

The formula for calculating the speed of a bullet is speed = distance / (time * cos(theta)), where distance is the horizontal distance traveled by the bullet, time is the total time of flight, and theta is the angle of elevation of the bullet's trajectory.

2. How do you derive the formula for the speed of a bullet?

The formula can be derived by using the basic principles of projectile motion, which state that the horizontal and vertical components of a projectile's motion are independent of each other. By considering the horizontal motion of the bullet and applying the cosine function to the angle of elevation, we can determine the bullet's velocity.

3. Does the mass of the bullet affect its speed?

Yes, the mass of the bullet does affect its speed. However, in the given formula, the mass of the bullet is not taken into account. This is because the formula assumes that the bullet has a constant mass and that there is no air resistance acting on the bullet.

4. Can this formula be used for all types of bullets?

Yes, this formula can be used for all types of bullets as long as the assumptions of constant mass and no air resistance hold true. However, it should be noted that the angle of elevation and distance traveled may vary for different types of bullets, which would affect the speed calculated.

5. How accurate is this formula in determining the speed of a bullet?

The accuracy of this formula depends on the accuracy of the measured values for distance, time, and angle of elevation. If these values are measured precisely, then the calculated speed will be accurate. However, factors such as air resistance, wind speed, and temperature can also affect the actual speed of a bullet, which may result in a slight discrepancy between the calculated and actual speed.

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